Tool to calculate the birthday paradox problem in probabilities. How many people are necessary to have a 50% chance that 2 of them share the same birthday.

Birthday Probabilities - dCode

Tag(s) : Statistics, Fun/Miscellaneous

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The birthday paradox is a mathematical problem put forward by Von Mises. It answers the question: what is the minimum number $ N $ of people in a group so that there is a 50% chance that at least 2 people share the same birthday (day-month couple). The answer is $ N = 23 $, which is quite counter-intuitive, most people estimate this number to be much larger, hence the paradox.

During the calculation of the birthdate paradox, it is supposed that births are equally distributed over the days of a year (it is not true in reality, but it's close).

In the following FAQ, a year has 365 days (leap years are ignored).

Probability is $ 1/365 \approx 0.0027 \approx 0.27\% $, indeed, 1 chance out of 365 to be born on a precise day in a year with 365 days.

__Example:__ A random person has a 0.27% chance of being born on April 1st (or any other day of the year).

__Example:__ Any average human has a 0.27% chance of being born on the same day as me/you.

NB: the complementary (opposite) probability of not being born on a certain day of the year is $ 1-1/365 = 364/365 \approx 0.9973 \approx 99.73\% $, indeed, there are 364 possible days out of 365.

__Example:__ A random person has a 99.73% chance of not being born on August 15 (or any other day of the year).

__Example:__ Any average human has a 99.73% chance of not being born on the same day as me/you.

By taking 2 people at random, and noting them A and B, this calculation amounts to asking the question *What is the probability that B was born on a certain day of the year?* this certain day being the birthday of birth of A. The probability is always $ 1/365 \approx 0.0027 \approx 0.27\% $

__Example:__ The probability that a mother was born on the same day as her child is 0.27%

__Example:__ The probability that a parent has a second child born on the same day (not the same year) as the first child is 0.27%

To calculate the odds for 2 people to be born on different days, take the opposite, so $ 1 - 1/365 = 364/365 \approx 0.9973 \approx 99.73\% $

Calculating this probability is equivalent to calculating the opposite of the probability that all people were born on a different day (because in this case at least 2 would be born on the same day).

For the smallest of the groups: N = 2. This calculation is explained above in *What is the probability that 2 people were born on the same day?* so $ P(N=2) \approx 0.27\% $. Note that this probability is the opposite of the probability that a person A was born on a different day from a person B and can therefore also be calculated $$ P(N=2) = 1 - (364/365) = 0.0027 = 0.27\% $$

For a larger group, N=3 composed of people A, B and C. There are therefore 364 chances out of 365 that B was not born on the same day as A and 363 chances out of 365 that C was not born the same days as A and B. Thus, $$ P(N=3) = 1 - (364/365) \times (363/365) \approx 0.82\% $$

In the same way, $$ P(N=4) = 1 - (364/365) \times (363/365) \times (362/365) \approx 1.64\% \\ P(N=5) = 1 - (364/365) \times \cdots \times (361/365) \approx 2.71\% $$

The general formula is $$ P(N=n) = 1 - \left(\frac{364}{365}\right) \times \left(\frac{363}{365}\right) \times \cdots \times \left(\frac{365-(n-1)}{365}\right) $$

A group of 23 people has a probability of approximately 0.5 or 50% (1 chance out of 2) that two people have a common birth date (day + month).

$$ P(N=23) = 1 - \left(\frac{364}{365}\right) \times \left(\frac{363}{365}\right) \times \cdots \times \left(\frac{343}{365}\right) \approx 0.5073 $$

The probability that no one shares a birthday is the opposite of the probability that there are (at least) 2 in common

$$ P(N=n) = \left(\frac{364}{365}\right) \times \left(\frac{363}{365}\right) \times \cdots \times \left(\frac{365-(n-1)}{365}\right) $$

By taking N people at random numbered from 1 to N, then by denoting D the date of birth of the first person, this amounts to calculating the probability that N-1 other people were born on date D. For person 2, the probability is $ P= 1/365 \approx 0.0027 \approx 0.27\% $, for person 3, same proba, $ P=1/365 $, etc. The probabilities are multiplied, i.e. the formula:

$$ P(X=N) = \left( \frac{1}{365} \right)^{N-1} $$

__Example:__ For $ N = 3 $ people, $ P(X=3) = 1/(365^2) \approx 0.0000075 \approx 0.00075\% $ or 1 chance out of (365*365)=133225

__Example:__ The probability that a mother and her 4 children (so 5 people) are born on the same day is 1 in 365^4

Calculate the opposite of the probability that the N people were born on another day.

$$ P(N=n) = 1 - (364/365)^(n-1) $$

__Example:__ $ P(N=3) = 1 - (364/365)^2 \approx 0,0055 \approx 0.55\% $

For a probability of 50%, a minimum of 253 people will be needed that the probability that 2 were born on a specific day is approximately 1/2.

__Example:__ $ P(N=253) = 1 - (364/365)^252 \approx 0.499105 \approx 50\% $

NB: The complementary probability that all N people are not born on a given date is P = (364/365)^N

366 are needed, ie, every day of the year plus one are needed to be sure that at least a couple of two people share the same birthday.

The answer would be 367 taking leap years into account.

With the hypothesis of a world population of 8 billion people.

People born on February 29 represent statistically 0.06653% of the population or 5 million people and are therefore negligible.

There are 8000000000/365 or about 22 million people born on a given day+month.

Limiting to the USA (350 millions inhabitants), there are 350000000/365 or nearly 1 millions US people born on a given day+month date.

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*Birthday Probabilities* on dCode.fr [online website], retrieved on 2022-11-28,

- Birthday Paradox Calculator
- What is the birthday paradox? (Definition)
- What is the probability for a person to be born on a given day of the year?
- What are the odds for 2 people to be born the same day?
- What are the odds for 2 people in a group of N to be born the same day?
- What are the odds for no one in a group of N to be born the same day?
- What is the probability for N people to be born the same day?
- What is the probability of at least 2 people among N to be born on a given day?
- How many people are needed in a group to be sure that 2 share the same birthday?
- How many people are born the same day as myself?

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